 \documentclass[12pt]{article}
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\usepackage{subcaption}
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\graphicspath{{./Graphs/}}
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\newtheorem{definition}{Definition}
\newtheorem{claim}{Claim}
\newtheorem{example}{Example}
\newtheorem{proposition}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{remark}{Remark}
\newtheorem{corollary}{Corollary}
\newtheorem{implication}{Implication}
\newtheorem{assumption}{Assumption}
\newtheorem{property}{Property}
\newcommand{\bit}{\begin{itemize}}


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\usepackage{graphicx}
\usepackage{tikz}
\graphicspath{ {./Graphs/} }
\begin {document} 


\begin{figure}
\begin{center}
\begin{tikzpicture}
\begin{axis}[
    axis lines = middle,
    xlabel = $x_1$,
    ylabel = {$E[U_L(x_1)]$}, 
    y label style={at={(axis description cs:0.5,-0.1)},anchor=north},
    x label style={at={(axis description cs:1,1)},anchor=south},
    yticklabels={,,,,},
    xticklabels={,,,}
]
   \addplot[color=black,solid,thick,mark=*]
   coordinates {
        (-1/2,0)
        (1/2,0)
        (-0.97,0)
        };
        \node [below] at (axis cs:  -1/2,0) {$-x'$};
        \node [above] at (axis cs:  1/2,-9/4) {$x^{pos}_L$};
        \node [below] at (axis cs:  1/2,0) {$x'$};
\node [below] at (axis cs:  -0.97,0) {$x^m_L$};
    
\addplot [
    domain=-1.5:-1/2, 
    samples=100, 
    color=black,
]
{-(x+1)^2};
\addplot [
    domain=-1/2:0, 
    samples=100, 
    color=black,
]
{-(x+1)^2-(1+2*x)*(4/3)^2};

\addplot [
    domain=1/2:1.1, 
    samples=100, 
    color=black,
]
{-(x+1)^2};

\addplot [
    domain=0:1/2, 
    samples=100, 
    color=black,
]
{-(x+1)^2-(1-2*x)*(4/3)^2};
\end{axis}
\end{tikzpicture}
\caption{$L$'s expected utility as a function of first-period policy}
\end{center}
\end{figure}
\end{document} 